深埋隧洞围岩应力的精确解与近似解的对比分析

对不同断面形状的深埋隧洞进行了分析，比较了隧洞围岩应力解析解与通过当量半径方法得到的近似解之间的差别.首先，应用复变函数的基本理论，给出圆形、椭圆、矩形、直墙拱形等几种常见深埋隧洞围岩应力的解析表达式.其次，应用当量半径的折算形式，将其任意形状的边界转化为标准圆形断面，利用Lamé解答得到了各围岩应力分量.最后，考虑隧洞断面形状参数的变化，通过数值算例对精确解和近似解进行了比较，分析了当量半径折算形式的精确度.在此基础上，应用有限元方法验证了复变函数解析解的精确性，以椭圆、矩形和直墙拱形的复变函数解验证当量半径精确度.结果表明，当量半径的折算形式解答与精确解答之间相似程度与隧洞的断面形状和几何参数之间有着密切的关系.

Comparison Between Exact Solutions and Approximate Solutions of Deep Tunnels

Deep buried tunnels with different cross-section shapes were analyzed through comparison between the analytical solutions and the approximate solutions obtained by the equivalent radii for the surrounding rock stresses. Firstly, according to the basic theory of complex variable functions, the stress components were obtained and the analytical expressions for the surrounding rock stresses of deep tunnels with various section shapes including the circle, the ellipse, the rectangle and the straight-wall arch, were given. Secondly, with the converted forms of the equivalent radii, the section boundary was transformed into a standard circle and the surrounding rock stress components were calculated by means of the Lamé solution. Finally, in view of the variable parameters of the tunnel cross-section shapes, comparisons between the exact solutions and the approximate solutions were conducted based on numerical examples, and the accuracy of the converted equivalent radii was discussed. The accuracy of the analytic solutions of the complex variable functions was proved, and the accuracy of the equivalent radii was verified for different cross-section shapes. The results show that the similarity between the solutions of the converted forms and the exact solutions of the equivalent radii is closely related to the cross-section shapes of the tunnels and the geometrical parameters.